Caustic Curves and Dual Curves: Are They Different?

In summary, the caustic curve is the same as the curve that best known in physics, which is nothing but the envelope of the rays reflected from a curved surface.
  • #1
mdoni
3
0
Hello,
I'm investigating duality for plane curves, and I came across an 'original' interpretation of the Biduality theorem , that uses the notion of caustic curve. Because everything is still very obscure to me, I try to share the whole with you, in the hope that we can help to fix ideas.
Meanwhile, some definitions to introduce the argument:

Definition: Let [tex]\mathbb{P}^*[/tex] the projective plane dual of [tex]\mathbb{P}^2 (\mathbb{C}) [/tex]: every line of [tex]\mathbb{P}^2 (\mathbb{C})[/tex] identifies a point of [tex]\mathbb{P}^*[/tex] and, conversely, every line of[tex] \mathbb{P}^*[/tex] corresponds to a point of [tex] \mathbb{P}^2 (\mathbb{C})[/tex]. Given a curve [tex]C \subset \mathbb{P}^2 (\mathbb {C}) [/tex], we consider the totality of the tangents to [tex]C[/tex]: it is a new curve in [tex] \mathbb{P}^* [/tex], the so-called dual curve [tex] C^*[/tex].

Biduality Theorem: For any projective curve [tex]C \subset \mathbb{P}^2 (\mathbb {C}) [/tex] we have [tex](C^*)^* = C[/tex]. Moreover, if [tex]p[/tex] is a simple point of [tex] C [/tex] and [tex]h[/tex] is a simple point of [tex]C^*[/tex], then [tex]h[/tex] is tangent to [tex]C[/tex] in [tex]p[/tex] if and only if [tex]p[/tex], considered as a straight line in [tex]\mathbb{P}^*[/tex] is tangent to [tex]C^*[/tex] in [tex]h[/tex].

Now, from the book Discriminants, Resultants, and Multidimensional Determinants (http://books.google.com/books?id=2z...resnum=1&ved=0CBUQ6AEwAA#v=onepage&q&f=false"), reported verbatim:

To give an intuitive sense of Biduality Theorem in [tex] C \subset {\mathbb{P}^2} (\mathbb {C})[/tex], we express the notion of tangency in the dual projective plane [tex] \mathbb{P}^* [/tex] in terms of the original plane [tex]\mathbb{P}^2(\mathbb{C}) [/tex]. By definition, a tangent to a curve at some point is the line that contains this point and that is infinitely close to the curve near this point.
In our situation, a point of [tex] \mathbb {P}^*[/tex] is a line [tex]l \subset \mathbb{P}^2 [/tex]. A curve [tex] C [/tex] in [tex]\mathbb{P}^*[/tex] is a 1-parameter family of lines in [tex] C \subset \mathbb {P}^2 (\mathbb{C}) [/tex]. A line in [tex]\mathbb{P}^*[/tex] is a pencil [tex]P^*[/tex] of all the lines in [tex]C \subset \mathbb{P}^2 (\mathbb{C}) [/tex] for a given point [tex] P [/tex] to [tex] C \subset \mathbb{P}^2 (\mathbb{C}) [/tex]. The condition that [tex] P^* [/tex] is tangent to [tex] C [/tex] in [tex] l [/tex] means that the line [tex] l [/tex] is a member of the family [tex] C [/tex], a point [tex] P [/tex] lies on [tex] l [/tex] and other lines of [tex] C [/tex] in the vicinity of [tex]l[/tex] are infinitely close to the pencil [tex] P^*[/tex]. This is usually expressed by saying that [tex] P [/tex] is a caustic point for the family of lines [tex] C [/tex].
One can imagine that a beam of light of a certain intensity is coming along each line of [tex] C [/tex]. Then the total brightness of the incoming light in an arbitrary small neighborhood of a caustic point [tex] P [/tex] is infinite, although there is only a ray (line of [tex] C [/tex]) that meets the point [ tex] P [/tex] itself. The set of all caustic points of the family of lines is usually called the caustic curve [/ b] of [tex]C[/tex]. This is nothing but the dual projective curve [tex] C^* \subset \mathbb{P}^2 [/tex].
Then the Biduality theorem states that any curve is the caustic of the family of its tangent lines (envelope of tangents). This is intuitively obvious.
The "dual" form of this theorem is less obvious: it means that every 1-parameter family in [tex] C [/tex] of lines in [tex] \mathbb{P}^2 (\mathbb{C}) [/tex] is the tangent line to any curve in [tex] \mathbb{P}^2 (\mathbb{C}) [/tex] and this curve is the caustic of [tex] C [/tex]. An example of a 1-parameter family of straight lines, which is not derived a priori as tangent lines to some curve is given by the reflection of a beam of parallel light in a curved mirror.

So now I ask: the caustic curve we are talking about is the same as the curve that best known in physics is nothing but the envelope of the rays reflected from a curved surface, and coming from a light source? Or maybe the dual of a curve is nothing but the caustic of the curve, caustic in the 'physical' sense given above?
And then, why should it be 'intuitively obvious' that each curve coincides with the caustic of its tangent lines (because what is gathered, the text defines the caustic curve from a family of 1-parameter lines , but the fact that these result in a curve as their envelope is a different kettle of fish)?
Finally, the dual form of the above consideration seems to me sincerely as obvious: trivially, every 1-parameter family of straight lines are tangents to any curve. What's wrong?
 
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  • #2
Sorry, how could I give the correct Tex-view to my post? I'm afraid, sorry.
 
  • #3
mdoni said:
Definition: Let [itex]\mathbb{P}^*[/itex] the projective plane dual of [itex]\mathbb{P}^2 (\mathbb{C}) [/itex]: every line of [itex]\mathbb{P}^2 (\mathbb{C})[/itex] identifies a point of [itex]\mathbb{P}^*[/itex] and, conversely, every line of[itex] \mathbb{P}^*[/itex] corresponds to a point of [itex] \mathbb{P}^2 (\mathbb{C})[/itex]. Given a curve [itex]C \subset \mathbb{P}^2 (\mathbb {C}) [/itex], we consider the totality of the tangents to [itex]C[/itex]: it is a new curve in [itex] \mathbb{P}^* [/itex], the so-called dual curve [itex] C^*[/itex].

Biduality Theorem: For any projective curve [itex]C \subset \mathbb{P}^2 (\mathbb {C}) [/itex] we have [itex](C^*)^* = C[/itex]. Moreover, if [itex]p[/itex] is a simple point of [itex] C [/itex] and [itex]h[/itex] is a simple point of [itex]C^*[/itex], then [itex]h[/itex] is tangent to [itex]C[/itex] in [itex]p[/itex] if and only if [itex]p[/itex], considered as a straight line in [itex]\mathbb{P}^*[/itex] is tangent to [itex]C^*[/itex] in [itex]h[/itex].

Now, from the book Discriminants, Resultants, and Multidimensional Determinants (http://books.google.com/books?id=2z...resnum=1&ved=0CBUQ6AEwAA#v=onepage&q&f=false"), reported verbatim:

To give an intuitive sense of Biduality Theorem in [itex] C \subset {\mathbb{P}^2} (\mathbb {C})[/itex], we express the notion of tangency in the dual projective plane [itex] \mathbb{P}^* [/itex] in terms of the original plane [itex]\mathbb{P}^2(\mathbb{C}) [/itex]. By definition, a tangent to a curve at some point is the line that contains this point and that is infinitely close to the curve near this point.
In our situation, a point of [itex] \mathbb {P}^*[/itex] is a line [itex]l \subset \mathbb{P}^2 [/itex]. A curve [itex] C [/itex] in [itex]\mathbb{P}^*[/itex] is a 1-parameter family of lines in [itex] C \subset \mathbb {P}^2 (\mathbb{C}) [/itex]. A line in [itex]\mathbb{P}^*[/itex] is a pencil [itex]P^*[/itex] of all the lines in [itex]C \subset \mathbb{P}^2 (\mathbb{C}) [/itex] for a given point [itex] P [/itex] to [itex] C \subset \mathbb{P}^2 (\mathbb{C}) [/itex]. The condition that [itex] P^* [/itex] is tangent to [itex] C [/itex] in [itex] l [/itex] means that the line [itex] l [/itex] is a member of the family [itex] C [/itex], a point [itex] P [/itex] lies on [itex] l [/itex] and other lines of [itex] C [/itex] in the vicinity of [itex]l[/itex] are infinitely close to the pencil [itex] P^*[/itex]. This is usually expressed by saying that [itex] P [/itex] is a caustic point for the family of lines [itex] C [/itex].
One can imagine that a beam of light of a certain intensity is coming along each line of [itex] C [/itex]. Then the total brightness of the incoming light in an arbitrary small neighborhood of a caustic point [itex] P [/itex] is infinite, although there is only a ray (line of [itex] C [/itex]) that meets the point [itex] P [/itex] itself. The set of all caustic points of the family of lines is usually called the caustic curve of [itex]C[/itex]. This is nothing but the dual projective curve [itex] C^* \subset \mathbb{P}^2 [/itex].
Then the Biduality theorem states that any curve is the caustic of the family of its tangent lines (envelope of tangents). This is intuitively obvious.
The "dual" form of this theorem is less obvious: it means that every 1-parameter family in [itex] C [/itex] of lines in [itex] \mathbb{P}^2 (\mathbb{C}) [/itex] is the tangent line to any curve in [itex] \mathbb{P}^2 (\mathbb{C}) [/itex] and this curve is the caustic of [itex] C [/itex]. An example of a 1-parameter family of straight lines, which is not derived a priori as tangent lines to some curve is given by the reflection of a beam of parallel light in a curved mirror.

So now I ask: the caustic curve we are talking about is the same as the curve that best known in physics is nothing but the envelope of the rays reflected from a curved surface, and coming from a light source? Or maybe the dual of a curve is nothing but the caustic of the curve, caustic in the 'physical' sense given above?
And then, why should it be 'intuitively obvious' that each curve coincides with the caustic of its tangent lines (because what is gathered, the text defines the caustic curve from a family of 1-parameter lines , but the fact that these result in a curve as their envelope is a different kettle of fish)?
Finally, the dual form of the above consideration seems to me sincerely as obvious: trivially, every 1-parameter family of straight lines are tangents to any curve. What's wrong?

The [noparse][itex][/itex][/noparse] tags give in-line LATEX.
 
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  • #4
Thanks.
No reply? :(
 
  • #5



I find this discussion on caustic curves and dual curves to be very interesting. It seems like you are trying to understand and interpret the Biduality Theorem in terms of caustic curves, which are commonly known in physics as the envelope of reflected rays from a curved surface. You are also questioning the intuition behind the theorem and its dual form.

First of all, let me clarify that the caustic curve we are discussing here is different from the physical caustic curve. The caustic curve in projective geometry is defined as the set of all caustic points of a 1-parameter family of lines, while the physical caustic curve is the envelope of reflected rays. However, the concept of caustic points and curves are related in the sense that the caustic curve in projective geometry is the dual of a curve, which is also known as the envelope of tangents.

Now, to address your question about why it is intuitively obvious that each curve coincides with the caustic of its tangent lines, we need to understand the concept of tangency in projective geometry. In this context, a tangent to a curve at a point is defined as the line that is infinitely close to the curve at that point. This means that the tangent line and the curve share the same point, but they are not identical. However, as we take more and more tangent lines at different points on the curve, the envelope of these lines will start to converge and eventually coincide with the curve itself. This is why it is intuitively obvious that a curve is the caustic of its tangent lines.

As for the dual form of the theorem, it is based on the duality principle in projective geometry, which states that there is a correspondence between points and lines in the projective plane. This means that for every point P on a curve C, there is a line in the dual plane that corresponds to it, and vice versa. Therefore, every 1-parameter family of lines in C can be seen as the tangent line to a point on the dual curve C^*. And since the dual curve is defined as the set of all caustic points of a 1-parameter family of lines, it follows that every 1-parameter family of lines in C is also the tangent line to the caustic of C. This is why the dual form of the theorem seems obvious.

In summary, caust
 

FAQ: Caustic Curves and Dual Curves: Are They Different?

1. What are caustic curves and dual curves?

Caustic curves and dual curves are mathematical concepts that describe the behavior of light rays as they reflect or refract off of a curved surface or through a curved medium. Caustic curves refer to the envelope of light rays reflected or refracted from a curved surface, while dual curves refer to the envelope of light rays that form a caustic curve when reflected or refracted again.

2. How are caustic curves and dual curves used in science?

Caustic curves and dual curves are used in various scientific fields such as optics, fluid dynamics, and computer graphics. In optics, they help understand the behavior of light in lenses and mirrors, while in fluid dynamics, they are used to study the flow of fluids around curved objects. In computer graphics, they are used to create realistic lighting effects.

3. What is the difference between caustic curves and dual curves?

The main difference between caustic curves and dual curves is the direction of light rays. Caustic curves refer to the envelope of light rays reflected or refracted from a curved surface, while dual curves refer to the envelope of light rays that form a caustic curve when reflected or refracted again.

4. Can caustic curves and dual curves occur in nature?

Yes, caustic curves and dual curves can occur in nature. Some examples include the ripples formed on the surface of a pond when a stone is thrown in, the patterns formed by light passing through a glass of water, and the shapes of rainbows formed by sunlight passing through water droplets in the atmosphere.

5. How are caustic curves and dual curves calculated and visualized?

Caustic curves and dual curves can be calculated using mathematical equations and algorithms. They can also be visualized using computer simulations and experiments. In computer graphics, they can be generated using ray tracing techniques. In experiments, they can be observed by shining a light source on a curved object and tracing the path of light rays using materials such as chalk dust or water droplets.

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